To solve the polynomial equation x³ – 64 = 0, we first recognize that this can be rewritten as x³ = 64. The next step is to find the cube roots of 64.
Since 64 = 4³, we know that one of the solutions is x = 4. However, a cubic equation can have up to three solutions, including complex ones. To find all the solutions, we can use the concept of cube roots in the complex plane.
In polar form, the number 64 can be expressed as 64 = 64(cos(0) + i sin(0)). The cube roots of a complex number can be found using the formula:
- r^{1/n} (cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)) for k = 0, 1, 2…
For our case, r = 64 and θ = 0, and since we want the cube roots, we set n = 3. Therefore, we calculate:
- Magnitude: 64^{1/3} = 4
- For k = 0: cos(0) + i sin(0) = 1 → Solution: 4
- For k = 1: cos(2π/3) + i sin(2π/3) = -1/2 + i(√3/2) → Solution: 4(-1/2 + i(√3/2)) = -2 + 2√3i
- For k = 2: cos(4π/3) + i sin(4π/3) = -1/2 – i(√3/2) → Solution: 4(-1/2 – i(√3/2)) = -2 – 2√3i
Thus, the three solutions to the equation x³ – 64 = 0 are:
- x = 4 (real solution)
- x = -2 + 2√3i (complex solution)
- x = -2 – 2√3i (complex solution)